It depends upon the kind of damage you are talking about, and the kind of coverage you have. If you are talking about damage the car thief caused to other people or property, then normally neither you nor your insurance will be responsible. If you're asking about damage to your car caused by the car thief, then your insurance coverage should pay for the repairs under your collision coverage, if you have it. Also, you will be responsible for any deductible under that coverage. If you do not have collision coverage you will either have to pay for the repairs to your own car yourself, or go after the car thief.
You should call your insurance company immediately to verify what coverages you have and how you can make a claim.
